Q.Which is the better investment, 3% per year compounded monthly or 3.1% per year compounded quarterly?

The key idea is to compare the Effective Annual Rate (EAR) for each option — the actual yearly return after accounting for compounding frequency.

Step 1: EAR formula

For a nominal rate $r$ compounded $n$ times per year:

Step 2: Compute EAR for 3% compounded monthly

$r = 0.03$, $n = 12$

So EAR ≈ 3.0416%.

Step 3: Compute EAR for 3.1% compounded quarterly

$r = 0.031$, $n = 4$

So EAR ≈ 3.138%.

Since 3.138% > 3.0416%, the second option yields a higher effective return.

Why we compare effective rates, not nominal ones

When interest is compounded more than once a year, the actual growth over a full year is higher than the stated nominal rate. This happens because each compounding period earns interest on the interest already credited. The Effective Annual Rate (EAR) is the single annual rate that would produce the same final amount after one year as the given compounding schedule. Comparing EARs is the only fair way to decide between two investments with different compounding frequencies.

Step-by-step comparison

1. Compute the EAR for 3% compounded monthly

• Nominal rate: $r = 0.03$
• Compounding periods per year: $n = 12$
• Monthly rate: $\frac{r}{n} = \frac{0.03}{12} = 0.0025$

The growth factor over one year is:

Now calculate:

Subtract 1 to get the EAR:

So a 3% nominal rate compounded monthly is equivalent to earning about 3.042% once per year.

2. Compute the EAR for 3.1% compounded quarterly

• Nominal rate: $r = 0.031$
• Compounding periods per year: $n = 4$
• Quarterly rate: $\frac{r}{n} = \frac{0.031}{4} = 0.00775$

The growth factor over one year is:

Calculate:

Subtract 1:

So a 3.1% nominal rate compounded quarterly is equivalent to about 3.136% per year.

3. Compare the two EARs

Since $3.136\% > 3.042\%$, the second investment yields a higher actual annual return.